Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

Abstract

. We consider a tower of generalized rook monoid algebras over the field C of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys–Murphy elements for generalized rook monoid algebras. Over an algebraically closed field k of positive characteristic p , utilizing Jucys–Murphy elements of rook monoid algebras, for 0 ≤ i ≤ p − 1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum G C of the Grothendieck groups of module categories over rook monoid algebras over k , these functors induce an action of the tensor product of the universal enveloping algebra U ( b sl p ( C )) and the monoid algebra C [ B ] of the bicyclic monoid B . Furthermore, we prove that G C is isomorphic to the tensor product of the basic representation of U ( b sl p ( C )) and the unique infinite-dimensional simple module over C [ B ], and also exhibit that G C is a bialgebra. Under some natural restrictions on the characteristic of k , we outline the corresponding result for generalized rook monoids.